An Algebraic Version of the Poincare Construction

Abstract

The Poincare construction in CR geometry allows us to estimate the dimension of the stabilizer in the Lie algebra of infinitesimal holomorphic automorphisms of the germ of a CR manifold by the dimension of the stabilizer in the corresponding algebra of the model surface of this germ. We give a negative answer to the following natural question: is there an algebraic Poincare construction, i.e., is it true that the stabilizer in the Lie algebra of automorphisms of the germ of a CR manifold is isomorphic to a Lie subalgebra of the stabilizer in the algebra of its model surface? We also give a negative answer to the corresponding question for the whole automorphisms algebra.